n^2+10n+1=0

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Solution for n^2+10n+1=0 equation:


Simplifying
n2 + 10n + 1 = 0

Reorder the terms:
1 + 10n + n2 = 0

Solving
1 + 10n + n2 = 0

Solving for variable 'n'.

Begin completing the square.

Move the constant term to the right:

Add '-1' to each side of the equation.
1 + 10n + -1 + n2 = 0 + -1

Reorder the terms:
1 + -1 + 10n + n2 = 0 + -1

Combine like terms: 1 + -1 = 0
0 + 10n + n2 = 0 + -1
10n + n2 = 0 + -1

Combine like terms: 0 + -1 = -1
10n + n2 = -1

The n term is 10n.  Take half its coefficient (5).
Square it (25) and add it to both sides.

Add '25' to each side of the equation.
10n + 25 + n2 = -1 + 25

Reorder the terms:
25 + 10n + n2 = -1 + 25

Combine like terms: -1 + 25 = 24
25 + 10n + n2 = 24

Factor a perfect square on the left side:
(n + 5)(n + 5) = 24

Calculate the square root of the right side: 4.898979486

Break this problem into two subproblems by setting 
(n + 5) equal to 4.898979486 and -4.898979486.


Subproblem 1
n + 5 = 4.898979486

Simplifying
n + 5 = 4.898979486

Reorder the terms:
5 + n = 4.898979486

Solving
5 + n = 4.898979486

Solving for variable 'n'.

Move all terms containing n to the left, all other terms to the right.

Add '-5' to each side of the equation.
5 + -5 + n = 4.898979486 + -5

Combine like terms: 5 + -5 = 0
0 + n = 4.898979486 + -5
n = 4.898979486 + -5

Combine like terms: 4.898979486 + -5 = -0.101020514
n = -0.101020514

Simplifying
n = -0.101020514


Subproblem 2
n + 5 = -4.898979486

Simplifying
n + 5 = -4.898979486

Reorder the terms:
5 + n = -4.898979486

Solving
5 + n = -4.898979486

Solving for variable 'n'.

Move all terms containing n to the left, all other terms to the right.

Add '-5' to each side of the equation.
5 + -5 + n = -4.898979486 + -5

Combine like terms: 5 + -5 = 0
0 + n = -4.898979486 + -5
n = -4.898979486 + -5

Combine like terms: -4.898979486 + -5 = -9.898979486
n = -9.898979486

Simplifying
n = -9.898979486


Solution
The solution to the problem is based on the solutions
from the subproblems.
n = {-0.101020514, -9.898979486}

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